Any suggestions for playing would be helpful. How does the on-line multiplayer work?
Basically, you go to the multiplayer lobby, join a game, set civilization, set difficulty to Prince, hit the checkmark for ready. If you are lucky, game will launch successfully. If not, do not get discouraged, just quit quickly and join a new game. Save/reload can fix an in-game problem as well, if game falls out of synch.
The main difference between multiplayer and single player game mechanics is simultaneous turns- everyone goes during the same turn. This complicates war with other players greatly, since it matters when you do an action in a specific turn, instead of just what turn you did that action on. It is very different to attack with a unit at the beginning of a turn [open to counter attack that turn] and at the end of a turn [not open to counter attack unless opponent is very fast/ready].
What happens when one player wants to leave when you are in the middle of playing?
An AI takes over that player's civilization. That player may also rejoin later, via "Join game" on Steam interface [seperate from Civ lobby!] and take over AI civilization again.
Can you save a multiplayer game?
Yes, multiplayer games auto-save. For some reason, they did not implement the usual saving mechanism into multiplayer. Therefore, to create a permanent copy of an auto-saved multiplayer game, find the auto-save file [via file search] on your computer and rename it whatever you want, and put it in the usual save game directory. If you do not do this, future autosaves will over-write the autosave you want to save. If re-loading immediately, just reload the autosave.